In Spot Metering (also called Partial Metering) mode the camera reads the light from a small spot in the centre of the frame. The light meter ignores everything else in the scene apart form a small, central spot:
This is useful in high contrast scenes when you want to correctly expose a specific area of the photo. In this example, the photographer wanted to ensure that the very centre of the photo was correctly exposed so that the lady stood out against the dark shadows. The spot meter has only read the light from the middle of the frame and has ignored the rest of the scene.
Spot Metering mode is the metering mode that requires the most thought and attention from you. This is because you have to be aware of what is in the centre of the frame every time you take a photo as this small portion of the scene is all that the light meter is reading from. If you forget you are using spot metering mode you may get some very unpredictable results due to the light meter reading the light from a specific area of the frame that you may not be concentrating on.